In order to simulate sylviculture with TROLL we need to implement a new sylviculture module inside TROLL model code. A first litterature review was completed by an interview with Laurent Descroix of the Office Nationale des Forêts. We discovered that rotten trees were not random and seemed to depend both on tree species and diameter. This document presents modelling of relation between rotten trees and their species and diameter.
In fact we have two different questions:
| N | Model |
|---|---|
| \(N\) | \(V_f \sim \mathcal{logN} (log[(\beta*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
| \(N_{p}\) | \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p)*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
| \(N_{s}\) | \(V_f \sim \mathcal{logN} (log[((\beta + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
| \(N_{p,s}\) | \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
We tested models N detailed in following tabs to find the better trade-off between:
Results are shown for each models in each model tabs and summarized in Conclusion tab.
\(N\): \(V_f \sim \mathcal{logN} (log[(\beta*dbh^2)*(1 - Pr*\rho)], \sigma)\)
\(P_{rotten} = (15.383*dbh^2)*[1-Pr*0.189]\)
\(N_{p}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p)*dbh^2)*(1 - Pr*\rho)], \sigma)\)
\(P_{rotten} = (15.259*dbh^2)*[1-Pr*0.228]\)
\(N_{s}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\)
\(P_{rotten} = (14.731*dbh^2)*[1-Pr*0.152]\)
\(N_{p,s}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\)
\(P_{rotten} = (14.782*dbh^2 + 1)*[1-Pr*0.209]\)
Table 3: Models summary.Adding random effect for plot and/or species (\(\beta_p\), \(\beta_s\)) increased likelihood and decreased model variance (\(\sigma\)). Consequently best model is \(N_{p,s}\). Stille we can try to improve this model by adding random effect for species and/or plot to the hyperparameter \(\rho\).
| L | Model |
|---|---|
| \(L\) | \(\rho = \theta *dbh^2\) |
| \(L_{p}\) | \(\rho = (\theta + \theta_p) *dbh^2\) |
| \(L_{s}\) | \(\rho = (\theta + \theta_s) *dbh^2\) |
| \(L_{p,s}\) | \(\rho = (\theta + \theta_p + \theta_s) *dbh^2\) |
\(L\): \(\rho = \theta *dbh^2\)
\(P_{rotten} = (13.715*dbh^2)*[1-Pr*(0.506*dbh^2)]\)
\(L_{p}\): \(\rho = (\theta + \theta_p) *dbh^2\)
\(P_{rotten} = (14.308*dbh^2)*[1-Pr*(0.502*dbh^2)]\)
\(L_{s}\): \(\rho = (\theta + \theta_s) *dbh^2\)
\(P_{rotten} = (13.56*dbh^2)*[1-Pr*(0.532*dbh^2)]\)
\(L_{p,s}\): \(\rho = (\theta + \theta_p + \theta_s) *dbh^2\)
\(P_{rotten} = (13.849*dbh^2)*[1-Pr*(0.561*dbh^2)]\)
Adding random effect for plot and/or species (\(\theta_p\), \(\theta_s\)) increased likelihood and decreased model variance (\(\sigma\)). Consequently best model is \(N_{p,s}\) associated to \(L_{p,s}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p + \beta_s)*dbh^2)*(1 - Pr*((\theta + \theta_p + \theta_s) *dbh^2))], \sigma)\)